scholarly journals The Bounds of Vertex Padmakar–Ivan Index on k-Trees

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 324 ◽  
Author(s):  
Shaohui Wang ◽  
Zehui Shao ◽  
Jia-Bao Liu ◽  
Bing Wei
Keyword(s):  
Molecular Structure ◽  
Topological Index ◽  
Upper Bounds ◽  
Extremal Graphs ◽  
Structure Descriptor

The Padmakar–Ivan ( P I ) index is a distance-based topological index and a molecular structure descriptor, which is the sum of the number of vertices over all edges u v of a graph such that these vertices are not equidistant from u and v. In this paper, we explore the results of P I -indices from trees to recursively clustered trees, the k-trees. Exact sharp upper bounds of PI indices on k-trees are obtained by the recursive relationships, and the corresponding extremal graphs are given. In addition, we determine the P I -values on some classes of k-trees and compare them, and our results extend and enrich some known conclusions.

scholarly journals Some Extremal Graphs with Respect to Sombor Index

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1202
Author(s):  
Kinkar Chandra Das ◽  
Yilun Shang
Keyword(s):  
Molecular Structure ◽  
Chromatic Number ◽  
Clique Number ◽  
Upper Bounds ◽  
Extremal Graphs ◽  
Lower And Upper Bounds ◽  
Graph Parameters ◽  
Structure Descriptor

Let G be a graph with set of vertices V(G)(|V(G)|=n) and edge set E(G). Very recently, a new degree-based molecular structure descriptor, called Sombor index is denoted by SO(G) and is defined as SO=SO(G)=∑vivj∈E(G)dG(vi)2+dG(vj)2, where dG(vi) is the degree of the vertex vi in G. In this paper we present some lower and upper bounds on the Sombor index of graph G in terms of graph parameters (clique number, chromatic number, number of pendant vertices, etc.) and characterize the extremal graphs.

scholarly journals Relating the ABC and harmonic indices

2014 ◽  
Vol 79 (5) ◽  
pp. 557-563 ◽  
Author(s):  
Ivan Gutman ◽  
Lingping Zhong ◽  
Kexiang Xu
Keyword(s):  
Molecular Structure ◽  
Topological Index ◽  
Molecular Graph ◽  
Vertex Degree ◽  
Harmonic Index ◽  
Abc Index ◽  
Atom Bond ◽  
Structure Descriptor

The atom-bond connectivity (ABC) index is a much-studied molecular structure descriptor, based on the degrees of the vertices of the molecular graph. Recently, another vertex-degree-based topological index - the harmonic index (H) - attracted attention and gained popularity. We show how ABC and H are related.

Estimating the Vertex PI Index

2010 ◽  
Vol 65 (3) ◽  
pp. 240-244 ◽  
Author(s):  
Kinkar Ch Das ◽  
Ivan Gutman
Keyword(s):  
Molecular Structure ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Pi Index ◽  
Structure Descriptor

The vertex PI index is a distance-based molecular structure descriptor, that recently found numerous chemical applications. Lower and upper bounds for PI are obtained, as well as results of Nordhaus-Gaddum type. Also a relation between the Szeged and vertex PI indices is established

Extended energy and its dependence on molecular structure

2017 ◽  
Vol 95 (5) ◽  
pp. 526-529 ◽  
Author(s):  
Ivan Gutman ◽  
Boris Furtula ◽  
Kinkar Ch. Das
Keyword(s):  
Molecular Structure ◽  
Electron Energy ◽  
Topological Index ◽  
Molecular Properties ◽  
Vertex Degree ◽  
Energy Formula ◽  
The Difference ◽  
Basic Characteristics ◽  
Ga Index ◽  
Structure Descriptor

The extended energy ([Formula: see text]) is a vertex degree based and spectrum-based molecular structure descriptor, shown to be well correlated with a variety of physicochemical molecular properties. We investigate the dependence of [Formula: see text] on molecular structure and establish its basic characteristics. In particular, we show how [Formula: see text] is related with the geometric–arithmetic (GA) topological index. Our main finding is that the difference between [Formula: see text] and the total π-electron energy is linearly proportional to the difference between the number of edges and the GA index.

scholarly journals Investigation of General Power Sum-Connectivity Index for Some Classes of Extremal Graphs

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Rui Cheng ◽  
Gohar Ali ◽  
Gul Rahmat ◽  
Muhammad Yasin Khan ◽  
Andrea Semanicova-Fenovcikova ◽  
...  
Keyword(s):  
Connected Graph ◽  
Topological Index ◽  
Upper Bounds ◽  
Connectivity Index ◽  
Extremal Graphs ◽  
Graph Invariant ◽  
Power Sum ◽  
General Power

In this work, we introduce a new topological index called a general power sum-connectivity index and we discuss this graph invariant for some classes of extremal graphs. This index is defined by Y α G = ∑ u v ∈ E G d u d u + d v d v α , where d u and d v represent the degree of vertices u and v , respectively, and α ≥ 1 . A connected graph G is called a k -generalized quasi-tree if there exists a subset V k ⊂ V G of cardinality k such that the graph G − V k is a tree but for any subset V k − 1 ⊂ V G of cardinality k − 1 , the graph G − V k − 1 is not a tree. In this work, we find a sharp lower and some sharp upper bounds for this new sum-connectivity index.

Estimating the Laplacian Energy-Like Molecular Structure Descriptor

2012 ◽  
Vol 67 (6-7) ◽  
pp. 403-406 ◽  
Author(s):  
Ivan Gutman
Keyword(s):  
Molecular Structure ◽  
Spanning Trees ◽  
Molecular Graph ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Laplacian Energy ◽  
Number Of Spanning Trees ◽  
Structure Descriptor ◽  
Better Than

Lower and upper bounds for the Laplacian energy-like (LEL) molecular structure descriptor are obtained, better than those previously known. These bonds are in terms of number of vertices and edges of the underlying molecular graph and of graph complexity (number of spanning trees)

Laplacian Energy of Digraphs and a Minimum Laplacian Energy Algorithm

2015 ◽  
Vol 26 (03) ◽  
pp. 367-380 ◽  
Author(s):  
Xingqin Qi ◽  
Edgar Fuller ◽  
Rong Luo ◽  
Guodong Guo ◽  
Cunquan Zhang
Keyword(s):  
Oriented Graph ◽  
Laplacian Matrix ◽  
Upper Bounds ◽  
Spectral Graph Theory ◽  
Extremal Graphs ◽  
Lower And Upper Bounds ◽  
Spectral Graph ◽  
Laplacian Energy ◽  
Energy Formula ◽  
Matrix Formula

In spectral graph theory, the Laplacian energy of undirected graphs has been studied extensively. However, there has been little work yet for digraphs. Recently, Perera and Mizoguchi (2010) introduced the directed Laplacian matrix [Formula: see text] and directed Laplacian energy [Formula: see text] using the second spectral moment of [Formula: see text] for a digraph [Formula: see text] with [Formula: see text] vertices, where [Formula: see text] is the diagonal out-degree matrix, and [Formula: see text] with [Formula: see text] whenever there is an arc [Formula: see text] from the vertex [Formula: see text] to the vertex [Formula: see text] and 0 otherwise. They studied the directed Laplacian energies of two special families of digraphs (simple digraphs and symmetric digraphs). In this paper, we extend the study of Laplacian energy for digraphs which allow both simple and symmetric arcs. We present lower and upper bounds for the Laplacian energy for such digraphs and also characterize the extremal graphs that attain the lower and upper bounds. We also present a polynomial algorithm to find an optimal orientation of a simple undirected graph such that the resulting oriented graph has the minimum Laplacian energy among all orientations. This solves an open problem proposed by Perera and Mizoguchi at 2010.

Sharp upper bounds of F-index among bicyclic graphs

2021 ◽  
pp. 2250055
Author(s):  
R. Khoeilar ◽  
A. Jahanbani ◽  
L. Shahbazi ◽  
J. Rodríguez
Keyword(s):  
Maximum Degree ◽  
Topological Index ◽  
Upper Bounds ◽  
Degree Of A Vertex ◽  
Bicyclic Graphs

The [Formula: see text]-index of a graph [Formula: see text], denoted by [Formula: see text], is defined as the sum of weights [Formula: see text] over all edges [Formula: see text] of [Formula: see text], where [Formula: see text] denotes the degree of a vertex [Formula: see text]. In this paper, we give sharp upper bounds of the [Formula: see text]-index (forgotten topological index) over bicyclic graphs, in terms of the order and maximum degree.

scholarly journals On Some New Neighborhood Degree-Based Indices for Some Oxide and Silicate Networks

J ◽  
2019 ◽  
Vol 2 (3) ◽  
pp. 384-409
Author(s):  
Sourav Mondal ◽  
Nilanjan De ◽  
Anita Pal
Keyword(s):  
Molecular Structure ◽  
Network Theory ◽  
Topological Index ◽  
Applied Mathematics ◽  
Reaction Network ◽  
Topological Indices ◽  
Zagreb Index ◽  
Second Zagreb Index ◽  
Real World Problems ◽  
Chemical Reaction Network Theory

Topological indices are numeric quantities that describes the topology of molecular structure in mathematical chemistry. An important area of applied mathematics is the chemical reaction network theory. Real-world problems can be modeled using this theory. Due to its worldwide applications, chemical networks have attracted researchers since their foundation. In this report, some silicate and oxide networks are studied, and exact expressions of some newly-developed neighborhood degree-based topological indices named as the neighborhood Zagreb index ( M N ), the neighborhood version of the forgotten topological index ( F N ), the modified neighborhood version of the forgotten topological index ( F N ∗ ), the neighborhood version of the second Zagreb index ( M 2 ∗ ), and neighborhood version of the hyper Zagreb index ( H M N ) are obtained for the aforementioned networks. In addition, a comparison among all the indices is shown graphically.

scholarly journals On the first Zagreb index and multiplicative Zagreb coindices of graphs

2016 ◽  
Vol 24 (1) ◽  
pp. 153-176 ◽  
Author(s):  
Kinkar Ch. Das ◽  
Nihat Akgunes ◽  
Muge Togan ◽  
Aysun Yurttas ◽  
I. Naci Cangul ◽  
...  
Keyword(s):  
Degree Sequence ◽  
Molecular Graph ◽  
Upper Bounds ◽  
Extremal Graphs ◽  
Lower And Upper Bounds ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
Irregularity Index ◽  
Vertex Set ◽  
The First Zagreb Index

AbstractFor a (molecular) graph G with vertex set V (G) and edge set E(G), the first Zagreb index of G is defined as, where dG(vi) is the degree of vertex vi in G. Recently Xu et al. introduced two graphical invariantsandnamed as first multiplicative Zagreb coindex and second multiplicative Zagreb coindex, respectively. The Narumi-Katayama index of a graph G, denoted by NK(G), is equal to the product of the degrees of the vertices of G, that is, NK(G) =. The irregularity index t(G) of G is defined as the number of distinct terms in the degree sequence of G. In this paper, we give some lower and upper bounds on the first Zagreb index M1(G) of graphs and trees in terms of number of vertices, irregularity index, maxi- mum degree, and characterize the extremal graphs. Moreover, we obtain some lower and upper bounds on the (first and second) multiplicative Zagreb coindices of graphs and characterize the extremal graphs. Finally, we present some relations between first Zagreb index and Narumi-Katayama index, and (first and second) multiplicative Zagreb index and coindices of graphs.

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